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Gentle Introduction to Particle Filtering

Sanjay Patil1 and Ryan Irwin2

Graduate research assistant1,

REU undergrad2

Human and Systems Engineering

URL: www.isip.msstate.edu/publications/seminars/msstate/2005/particle/

- Abstract

- Particle Filtering:
- Most conventional techniques for speech analysis are based on modeling signals as Gaussian Mixture Models in Hidden Markov Model based systems.
- To overcome the mismatched channel conditions, and/or significantly reduce the complexity of the models, Nonlinear approaches are expected to perform better than the conventional techniques.
- Particle filters, based on sequential Monte Carlo methods, is one such nonlinear methods.
- Particle filtering allows complete presentation of the posterior distribution of the states. Statistical estimates can be computed easily even in the presence of nonlinearities.

- Outline of Presentation

- Nonlinear Methods â€“ necessity
- Drawing Samples from a Probability distribution. (introduce â€˜Particleâ€™)
- Sequential Monte Carlo Methods â€“ necessity, different names â€“ bootstrap, condensation algorithm, survival of the fittest.
- Steps in particle filtering (explaining the algorithm â€“ block schematic)
- Actual example â€“ (along with all the steps)
- Brief review and applications for tracking
- As can be applied to Speaker Verification
- Demo

- Drawing samples from a probability distribution function

- Concept of samples and its weights

200 samples

- Take p(x)=Gamma(4,1)
- Generate some random samples
- Plot basic approximation to pdf
- Each sample is called as â€˜Particleâ€™

500 samples

5000 samples

- Particle filtering -

- Condensation Algorithm
- Survival of the fittest

- Different Names â€“
- Sequential Monte Carlo filters
- Bootstrap filters

General Problem Statement â€“ Filtering â€“ estimation of the states

- Tracking the state (parameters or hidden variables) as it evolves over time
- Sequentially arriving (noisy and non-Gaussian) observations
- Idea is to have best possible estimate of hidden variables

- Particle filtering algorithm continueâ€¦

General two-stage Framework

(Prediction-Update stages)

- Assume that pdf p(xk-1 | y1:k-1) is available at time k -1.
- Prediction stage:
- This is the prior of the state at time k ( without the information on measurement). Thus, it is the probability of the the state given only the previous measurements

- Update stage:
- This is posterior pdf from predicted prior pdf and newly available measurement.

- Particle filtering algorithm step-by-step (1)

- Particle filtering step-by-step (2)

- Particle filtering step-by-step (3)

- Particle filtering step-by-step (4)

- Particle filtering step-by-step (5)

- Particle filtering step-by-step (6)

- Particle filtering - visualization

- Drawing samples

- Predicting next state

- Updating this state

- What is THIS STEP???

- Resamplingâ€¦.

- Sampling Importance Resample algorithm (necessity)

- Applications:

- All the applications are mostly tracking applications in different formsâ€¦.
- Visual Tracking â€“ e.g. human motion (body parts)
- Prediction of (financial) time series â€“ e.g. mapping gold price, stocks
- Quality control in semiconductor industry
- Military Applications
- Target recognition from single or multiple images
- Guidance of missiles
- What is the application for IES NSF funded project â€“
- Time series estimation for speech signal (Java demo)
- Speaker Verification (details on next slide)

- Pattern Recognition Applet

- Java applet that gives a visual of algorithms implemented at IES
- Classification of Signals:
- PCA - Principle Component Analysis
- LDA - Linear Discrimination Analysis
- SVM - Support Vector Machines
- RVM - Relevance Vector Machines

- Tracking of Signals
- LP - Linear Prediction
- KF - Kalman Filtering
- PF â€“ Particle Filtering

- Pattern Classification

- Different data sets need to be differentiated without looking at all the data samples
- Classifications distinguishes between sets of data without the samples
- Algorithms separate data sets with a line of discrimination
- To have zero error the line of discrimination should completely separate the classes
- These patterns are easy to classify

- Pattern Classification

- Toroidals are not classified very successfully with a straight line
- Error should be around 50% because half of each class is separated
- A proper line of discrimination of a toroidal would be a circle enclosing only the inside set

- Signal Tracking

- The input signals are now time based with the x-axis representing time
- All the signal tracking algorithms are implemented with interpolated data
- The interpolation ensures that the input samples are at regular intervals
- Sampling is always done on regular intervals
- The linear prediction algorithm is a linear way to predict signals with no noise

- Signal Tracking

- The Kalman filter and particle filter are based on prediction of the states of the signal
- States are related to the observations through the state equation
- The particle filtering algorithm introduces process and measurement noise
- At each iteration possible states are given by the black points
- The average of the black points is where the overall state is predicted to be

- References:

- S. Haykin and E. Moulines, "From Kalman to Particle Filters," IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, Pennsylvania, USA, March 2005.
- M.W. Andrews, "Learning And Inference In Nonlinear State-Space Models," Gatsby Unit for Computational Neuroscience, University College, London, U.K., December 2004.
- P.M. Djuric, J.H. Kotecha, J. Zhang, Y. Huang, T. Ghirmai, M. Bugallo, and J. Miguez, "Particle Filtering," IEEE Magazine on Signal Processing, vol 20, no 5, pp. 19-38, September 2003.
- N. Arulampalam, S. Maskell, N. Gordan, and T. Clapp, "Tutorial On Particle Filters For Online Nonlinear/ Non-Gaussian Bayesian Tracking," IEEE Transactions on Signal Processing, vol. 50, no. 2, pp. 174-188, February 2002.
- R. van der Merve, N. de Freitas, A. Doucet, and E. Wan, "The Unscented Particle Filter," Technical Report CUED/F-INFENG/TR 380, Cambridge University Engineering Department, Cambridge University, U.K., August 2000.
- S. Gannot, and M. Moonen, "On The Application Of The Unscented Kalman Filter To Speech Processing," International Workshop on Acoustic Echo and Noise, Kyoto, Japan, pp 27-30, September 2003.
- J.P. Norton, and G.V. Veres, "Improvement Of The Particle Filter By Better Choice Of The Predicted Sample Set," 15th IFAC Triennial World Congress, Barcelona, Spain, July 2002.
- J. Vermaak, C. Andrieu, A. Doucet, and S.J. Godsill, "Particle Methods For Bayesian Modeling And Enhancement OfÂ Speech Signals," IEEE Transaction on Speech and Audio Processing, vol 10, no. 3, pp 173-185, MarchÂ 2002.